In the case of L1157-B1, previous observational work showed that the molecular gas emission arises from a region with a typical gaussian size (FWHM) in the range 20 – 25 [1]}, [2]}, [3]}. This is consistent with the distribution of the HCN \(J\) =3–2 emission (Fig. REF ), which appears extended between the protostar and the B1 shock position. The high-\(J\)  transitions of the HCN and HNC isotopologues lie in a frequency range for which the main-beam of the telescope is comparable to or smaller than the size of the shock region, as estimated from interferometric observations (see Tables REF –REF ). For these transitions, the emission can be considered as extended and the main-beam temperature becomes a satisfying approximation to the molecular line brightness temperature.
The case of the ground state transitions of the HCN and HNC isotopologues is more complicated for two reasons: first, the telescope beamwidth (HPBW) is comparable to the molecular emission size; second, the typical physical conditions
(\(n(\hbox{H${}_2$})\sim 10^5\hbox{\hspace{1.99997pt}cm$^{-3}$}\) ; \(T_{kin}\) = \(60\hbox{\hspace{1.99997pt}K}\) –\(90\hbox{\hspace{1.99997pt}K}\) ) are such than the rotational transitions present in the millimeter bands are subthermally excited. The \(J\) =1–0 excitation temperature is higher than that of the
\(J\) =3–2 line, as can be checked by a simple calculation using a radiative transfer code in the Large Velocity Gradient approximation, like Madex (Cernicharo et al. 2012). In the LTE approximation, the \(T_{ex}\)  of the \(J\) =1–0 line is underestimated by the fitting procedure, and the approximation is no longer very satisfying. However, the approximation of extended, or main-beam averaged, emission compensates for the previous effect, which mainly affects the ground state transitions, as explained above. We found that this approximation yields much better quality LTE fits. The source-averaged column density is then obtained by correcting the fitted (main-beam averaged) column density for the main-beam filling factor.
